More Electron Traps — Question 3

Question 3

The harmonic position operator permits only neighboring-level dipole transitions

A one-dimensional harmonic trap has $V(x)=m\omega^2x^2/2$ and $E_n=(n+1/2)\hbar\omega$, $n=0,1,2,\ldots$. Use $x=\sqrt{\hbar/(2m\omega)}(a+a^\dagger)$, $a|n\rangle=\sqrt n|n-1\rangle$ and $a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle$. A weak electric field couples linearly to x.

1. Derive $\langle r|x|n\rangle$ and identify the allowed index changes.

2. Compare $|x_{21}|^2$ with $|x_{10}|^2$ and assess a direct $0\to2$ electric-dipole transition.

Question 3 - Solution

The harmonic position operator permits only neighboring-level dipole transitions

(a) Method and calculation

$$\langle r|x|n\rangle=\sqrt{\frac{\hbar}{2m\omega}}\left(\sqrt n\,\delta_{r,n-1}+\sqrt{n+1}\,\delta_{r,n+1}\right).$$$$\boxed{\Delta n=\pm1}\quad\text{for nonzero position matrix elements}.$$

(b) Interpretation and further calculation

$$\boxed{|x_{21}|^2/|x_{10}|^2=2}.$$$$\boxed{x_{20}=0}\Rightarrow\boxed{0\to2\text{ absent in first-order dipole coupling}}.$$

Physical check

The ratio compares matrix-element strengths, not automatically total transition rates in different radiation environments. Multiphoton or nonlinear coupling is outside this model.